Homework 5

1. At $t=0$, a 1D harmonic oscillator is in a linear combination of the energy eigenstates
   
   $$\psi=\sqrt{2\over 5}u_3+i\sqrt{3\over 5}u_4$$
   
   
   Find the expected value of $p$ as a function of time using operator methods.

2. Evaluate the ``uncertainty'' in $x$ for the 1D HO ground state $\sqrt{\langle u_0\vert(x-\bar{x})^2\vert u_0\rangle}$ where $\bar{x}=\langle u_0\vert x\vert u_0\rangle$. Similarly, evaluate the uncertainty in $p$ for the ground state. What is the product $\Delta p\Delta x$? Now do the same for the first excited state. What is the product $\Delta p\Delta x$ for this state?

3. An operator is Unitary if $UU^\dagger=U^\dagger U=1$. Prove that a unitary operator preserves inner products, that is $\langle U\psi\vert U\phi\rangle=\langle\psi\vert\phi\rangle$. Show that if the states $\vert u_i\rangle$ are orthonormal, that the states $U\vert u_i\rangle$ are also orthonormal. Show that if the states $\vert u_i\rangle$ form a complete set, then the states $U\vert u_i\rangle$ also form a complete set.

4. Show at if an operator $H$ is hermitian, then the operator $e^{iH}$ is unitary.

5. Calculate $\langle u_i\vert x\vert u_j\rangle$ and $\langle u_i\vert p\vert u_j\rangle$.

6. Calculate $\langle u_i\vert xp\vert u_j\rangle$ by direct calculation. Now calculate the same thing using $\sum\limits_k\langle u_i\vert x\vert u_k\rangle\langle u_k\vert p\vert u_j\rangle$.

7. If $h(A^\dagger)$ is a polynomial in the operator $A^\dagger$, show that $Ah(A^\dagger)u_0={d h(A^\dagger)\over d A^\dagger}u_0$. As a result of this, note that since any energy eigenstate can be written as a series of raising operators times the ground state, we can represent $A$ by ${d\over d A^\dagger}$.

8. At $t=0$ a particle is in the one dimensional Harmonic Oscillator state $\psi(t=0)={1\over\sqrt{2}}(u_0+u_1)$.
   • Compute the expected value of $x$ as a function of time using $A$ and $A^\dagger$ in the Schrodinger picture.
   • Now do the same in the Heisenberg picture.